Subgroups — #27270683

Order 13568.

292 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274867	C₂	1	1	yes	maximal, characteristic
2	#20274832	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1831	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	1	no	—
5	C₂×C₄⋊#88322	—	2	1	no	—
6	C₂×C₄⋊#88322	—	2	1	no	—
7	C₂×C₄⋊#88322	C₄	1	2	yes	—
8	#16810378	—	2	1	no	—
9	#16810378	C₄	1	2	yes	—
10	C₂×C₄⋊C₈×C₅₃	—	2	1	no	—
11	C₄³×C₅₃	C₂²	1	1	yes	characteristic
12	C₅₃×#336	—	2	1	no	—
13	C₅₃×#336	C₄	1	2	yes	—
14	C₂²×#88322	—	2	2	no	—
15	C₂²×#88322	—	2	2	no	—
16	C₄⋊#88322	—	4	1	no	—
17	C₄⋊#88322	—	4	1	no	—
18	C₄⋊#88322	—	4	1	no	—
19	C₄⋊#88322	—	4	1	no	—
20	C₄⋊#88322	—	4	1	no	—
21	C₄⋊#88322	—	4	1	no	—
22	C₄⋊#88322	—	4	2	no	—
23	C₂²×C₈×C₅₃	—	2	2	no	—
24	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	Q₈	1	1	yes	characteristic
31	C₄⋊C₈×C₅₃	—	4	1	no	—
32	C₄⋊C₈×C₅₃	—	4	1	no	—
33	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	—	4	2	no	—
43	C₄²×C₅₃	—	2	1	no	—
44	C₄²×C₅₃	—	2	1	no	—
45	C₄²×C₅₃	—	2	2	no	—
46	C₄²×C₅₃	—	4	1	no	—
47	C₄²×C₅₃	—	4	1	no	—
48	C₄²×C₅₃	—	4	1	no	—
49	C₄²×C₅₃	D₈	1	1	yes	characteristic
50	C₄²×C₅₃	D₈	1	1	yes	characteristic
51	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
52	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
53	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
54	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
55	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
56	C₄²×C₅₃	Q₁₆	1	1	yes	characteristic
57	C₂×#88322	—	4	2	no	—
58	C₂×#88322	—	4	2	no	—
59	C₂×#88322	—	4	2	no	—
60	C₂×#88322	—	4	2	no	—
61	C₂×#88322	—	4	2	no	—
62	C₂×#88322	—	4	2	no	—
63	C₂³×C₅₃	#95	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	—	2	1	no	—
77	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
79	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
81	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
82	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
87	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#108	1	2	yes	—
92	C₈×C₅₃	—	8	2	no	—
93	#88322	—	8	2	no	—
94	#88322	—	8	2	no	—
95	#7040	—	53	1	no	maximal, Sylow
96	C₂²×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
97	C₂²×C₅₃	#340	1	1	yes	characteristic
98	C₂²×C₅₃	#340	1	1	yes	characteristic
99	C₂²×C₅₃	#341	1	1	yes	characteristic
100	C₂²×C₅₃	#343	1	1	yes	characteristic
101	C₂²×C₅₃	#344	1	1	yes	characteristic
102	C₂²×C₅₃	#344	1	1	yes	characteristic
103	C₄×C₅₃	—	2	1	no	—
104	C₄×C₅₃	—	2	1	no	—
105	C₄×C₅₃	—	2	1	no	—
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	—	4	1	no	—
113	C₄×C₅₃	#331	1	1	yes	characteristic
114	C₄×C₅₃	#332	1	1	yes	characteristic
115	C₄×C₅₃	#332	1	1	yes	characteristic
116	C₄×C₅₃	#333	1	1	yes	characteristic
117	#1842	—	53	1	no	—
118	#1831	—	53	1	no	—
119	#1831	D₅₃	1	1	yes	characteristic
120	C₂×C₅₃	#1286	1	1	yes	characteristic
121	C₂×C₅₃	#1286	1	1	yes	characteristic
122	C₂×C₅₃	#1290	1	1	yes	characteristic
123	C₂×C₅₃	#1296	1	1	yes	characteristic
124	C₂×C₅₃	#1301	1	1	yes	characteristic
125	C₂×C₅₃	#1301	1	1	yes	characteristic
126	C₂×C₅₃	#1307	1	1	yes	characteristic
127	C₂×C₄⋊C₈	—	2	1	no	—
128	C₂×C₄⋊C₈	—	53	2	no	—
129	C₂×C₄⋊C₈	—	106	1	no	—
130	C₂×C₄⋊C₈	—	106	1	no	—
131	C₂×C₄⋊C₈	—	106	1	no	—
132	C₄³	C₂×D₅₃	1	1	yes	characteristic
133	#336	—	2	1	no	—
134	#336	—	53	2	no	—
135	#336	—	106	1	no	—
136	#336	Dic₅₃	1	2	yes	—
137	C₅₃	#7040	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	106	2	no	—
140	C₂²×C₈	—	106	2	no	—
141	C₂×C₄²	Dic₁₀₆	1	1	yes	characteristic
142	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
143	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
144	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
147	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₄⋊C₈	—	4	1	no	—
149	C₄⋊C₈	—	4	1	no	—
150	C₄⋊C₈	—	212	2	no	—
151	C₄⋊C₈	—	212	1	no	—
152	C₄⋊C₈	—	212	1	no	—
153	C₄⋊C₈	—	212	1	no	—
154	C₄⋊C₈	—	212	1	no	—
155	C₄⋊C₈	—	212	1	no	—
156	C₄⋊C₈	—	212	1	no	—
157	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₄	#11719919	1	1	yes	characteristic
159	C₂²×C₄	#11719919	1	1	yes	characteristic
160	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
161	C₂²×C₄	#11719921	1	1	yes	characteristic
162	C₂²×C₄	#11719921	1	1	yes	characteristic
163	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	212	2	no	—
168	C₂×C₈	—	212	2	no	—
169	C₂×C₈	—	212	2	no	—
170	C₂×C₈	—	212	2	no	—
171	C₂×C₈	—	212	2	no	—
172	C₂×C₈	—	212	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	4	1	no	—
177	C₄²	—	4	1	no	—
178	C₄²	—	4	1	no	—
179	C₄²	#11719913	1	1	yes	characteristic
180	C₄²	#11719913	1	1	yes	characteristic
181	C₄²	D₄₂₄	1	1	yes	characteristic
182	C₄²	Dic₂₁₂	1	1	yes	characteristic
183	C₄²	#11719922	1	1	yes	characteristic
184	C₄²	#11719923	1	1	yes	characteristic
185	C₄²	#11719924	1	1	yes	characteristic
186	C₄²	#11719925	1	1	yes	characteristic
187	C₂³	#13133124	1	1	yes	characteristic
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	#13133133	1	1	yes	characteristic
202	C₂×C₄	#13133135	1	1	yes	characteristic
203	C₂×C₄	#13133137	1	1	yes	characteristic
204	C₂×C₄	#13133138	1	1	yes	characteristic
205	C₂×C₄	#13133139	1	1	yes	characteristic
206	C₂×C₄	#13133139	1	1	yes	characteristic
207	C₂×C₄	#13133140	1	1	yes	characteristic
208	C₂×C₄	#13133141	1	1	yes	characteristic
209	C₂×C₄	#13133142	1	1	yes	characteristic
210	C₂×C₄	#13133142	1	1	yes	characteristic
211	C₂×C₄	#13133148	1	1	yes	characteristic
212	C₂×C₄	#13133149	1	1	yes	characteristic
213	C₂×C₄	#13133150	1	1	yes	characteristic
214	C₂×C₄	#13133151	1	1	yes	characteristic
215	C₂×C₄	#13133152	1	2	yes	—
216	C₈	—	8	2	no	—
217	C₈	—	424	2	no	—
218	C₈	—	424	2	no	—
219	C₂²	#16810379	1	1	yes	characteristic
220	C₂²	#16810385	1	1	yes	characteristic
221	C₂²	#16810386	1	1	yes	characteristic
222	C₂²	#16810387	1	1	yes	characteristic
223	C₂²	#16810391	1	1	yes	characteristic
224	C₂²	#16810393	1	1	yes	characteristic
225	C₂²	#16810394	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	1	no	—
236	C₄	#16810366	1	1	yes	characteristic
237	C₄	#16810368	1	1	yes	characteristic
238	C₄	#16810370	1	1	yes	characteristic
239	C₄	#16810371	1	1	yes	characteristic
240	C₂	#20272573	1	1	yes	characteristic
241	C₂	#20272574	1	1	yes	characteristic
242	C₂	#20272584	1	1	yes	characteristic
243	C₂	#20272599	1	1	yes	characteristic
244	C₂	#20272610	1	1	yes	characteristic
245	C₂	#20272611	1	1	yes	characteristic
246	C₂	#20272625	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133124
commutator	C₂²×C₄×C₅₃	C₄²
Frattini	C₄³	C₂×D₅₃
Fitting	C₅₃×#1831	C₂
socle, prime radical	C₂³×C₅₃	#95
cosocle	C₄³×C₅₃	C₂²
hypercenter	#1831	D₅₃
hypocenter	C₅₃	#7040
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1831

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
137. C₅₃	#7040	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274867	2
#20274832	2
C₅₃×#1831	2
#7040	53